Abstract
We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[−β|t−s|−p] wheret ands are the times at which the common site is visited andp is a parameter. We prove that ifp<1 and β is sufficiently large, then the walk behaves ballistically, i.e., the distance to the end of the walk grows linearly with the number of steps in the walk. We also give a heuristic argument that ifp>3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation.
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References
E. Bolthausen, On self-repellent one dimensional random walks,Prob. Theory Related Fields 86:423 (1990).
D. Brydges, S. Evans, and J. Imbrie, Self-avoiding walk on a hierarchical lattice in four dimensions,Ann. Prob. 20:82 (1992).
D. Brydges and G. Slade, The diffusive phase of a model of self-interacting walks, preprint (1994).
D. Brydges and T. Spencer, Self-avoiding walk in 5 or more dimensions,Commun. Math. Phys. 97:125 (1985).
S. Caracciolo, G. Parisi, and A. Pelissetto, Random walks with short-range interaction and mean-field behavior, preprint.
A. Greven and F. den Hollander, A variational characterization of the speed of a one-dimensional self-repellent random walk,Ann. Appl. Prob. 3:1067 (1993).
T. Hara and G. Slade, Self-avoiding walk in five or more dimensions I. The critical behavior,Commun. Math. Phys. 147:101 (1992).
T. Hara and G. Slade, The lace expansion for self-avoiding walk in five or more dimensions,Rev. Math. Phys. 4:235 (1992).
D. Iagolnitzer and J. Magnen, Polymers in a weak random potential in dimension four: Rigorous renormalization group analysis,Commun. Math. Phys., to appear.
N. Madras and G. Slade,The Self-Avoiding Walk (Birkhäuser, Basel, 1993).
G. Slade, The diffusion of self-avoiding random walk in high dimensions,Commun. Math. Phys. 110:661 (1987).
H. Zoladek, One-dimensional random walk with self-interaction,J. Stat. Phys. 47:543 (1987).
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Kennedy, T. Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty. J Stat Phys 77, 565–579 (1994). https://doi.org/10.1007/BF02179450
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DOI: https://doi.org/10.1007/BF02179450